On the Persistence of Lower Dimensional Invariant Tori under Quasi-Periodic Perturbations

Abstract

In this work w e c o n s i d er time d ependent quasiperiodic perturbations of autonomous Hamiltonian systems. We focus on the e ect that t his kind o f p e r t urbations has on lower dimensional invariant t ori. Our results s h ow t hat, under standard conditions of analyticity, nondegeneracy and nonresonance, most of these tori survive, adding the frequencies of the p e r t urbation to t he o n es they already have.The paper also contains estimates on the amount of surviving t ori. The w orst situation happens when the initial tori are normally elliptic. In this case, a torus (identi ed by t he v ector of intrinsic frequencies) can be continued with respect to a p e r t urbative p a r a m eter " 2 0 " 0 ], except for a set of " of measure exponentially small with " 0 . I n c a s e t hat " is xed (and su ciently small), we prove t he existence of invariant t ori for every vector of frequencies close to t he o n e o f t he initial torus, except for a set of frequencies of measure exponentially small with t he d i s t ance to the u nperturbed torus. As a particular case, if the p e r t urbation is autonomous, these results also give t he s a m e k i n d o f e s t imates on the m easure of destroyed tori.Finally, t hese results are applied to s o m e problems of celestial mechanics, in order to h elp in the d escripti o n o f t he p h ase space of some concrete m o d els.jorba@ma1.upc.es y jordi@tere.upc.es 2

A New Approach to the Parameterization Method for Lagrangian Tori of Hamiltonian Systems

“…A natural way to measure this distance is through the norm of the matrixfunction Ω (see (2)). As shown in [8], the size of Ω can be related to the size of the invariance error e of τ (see (17)) through the Lie derivative L ω Ω (see (21)). Proceeding in this way, we end up with an estimate of the form Ω ρ−2δ = O(γ −1 δ −ν−1 e ρ ), which involves a division by γ that we want to avoid.…”

“…Equation (3) means that the correspondence t ∈ R → τ (ωt + θ 0 ) defines a quasi-periodic trajectory of h, ∀θ 0 ∈ T r . Equation (3) also implies that L ω Ω = 0 (see (17) and (21)). Since Ω θ = 0 by definition 2.1, we have that Ω = 0 (see (5)).…”

“…If h is a ε-small perturbation of a Hamiltonian for which we know the parameterization τ of an invariant torus, then the size of e j is of O(ε), j = 1, 2, 3 (in fact, we can set e 2 = 0). If h is not a natural perturbation of a simpler system, but we know the size of the invariance error e associated to the parameterization τ of a quasi-torus of h (see (17)), then we can relate each e j to e (see (36)). However, bounding e 1 and e 2 in terms of e involves the resolution of a small divisors equation, which means an estimate of the form O( e ρ /γ).…”

“…To control the measure of the portion of the phase space filled by these invariant tori, it is enough to show that the dependence ω → τ * ω is Lipschitz (e.g., see the elegant proof of the classical KAM theorem performed in [29]). To carry out the explict control of the Lipschitz constant of ω → τ * ω , we may use analogous strategies as those in [16,17] for proving existence of families of lower dimensional tori of Hamiltonian systems, of dimension s < r, using ω ∈ R s as a parameter.…”

“…Firstly, we consider the invariance error e of τ (see (17)) and the matrix representation Ω of the pullback by τ of the symplectic form. Both expressions appear explicitly in some computations, so we need to relate them to e 1 , e 2 , and e 3 .…”